Algebraic and Hermitian K-theory of K-rings

The main purpose of the present article is to establish the real case of “Karoubi’s Conjecture” [20] in algebraic K-theory. The complex case was proved in 1990-91 (cf. [32] for the C∗-algebraic form of the conjecture, and [38] for both the Banach and C∗-algebraic forms). Compared to the case of complex algebras, the real case poses additional difficulties. This is due to the fact that topological K-theory of real Banach algebras has period 8 instead of 2. The method we employ to overcome these difficulties can also be used for complex algebras, and provides some simplifications to the original proofs. We also establish a natural analog of Karoubi’s Conjecture for Hermitian K-theory. Introduction Let A be a complex C∗-algebra and K = K(H) be the ideal of compact operators on a separable complex Hilbert space H. One of us conjectured around 1977 that the natural comparison map between the algebraic and topological K-groups en : Kn(K⊗̄A) −→ K n (K⊗̄A) ' K top n (A) is an isomorphism for a suitably completed tensor product ⊗̄. The conjecture was announced in [20] where accidentally only K⊗̂π A was mentioned, and it was proved there for n ≤ 0. In [23] the conjecture was established for n ≤ 2 and K⊗̄A having the meaning of the C∗-algebra completion of the algebraic tensor product (in view of nuclearity of K, there is only one such completion). The C∗-algebraic form of the conjecture was established for all n ∈ Z in 1990 [31], [32]. A year later the conjecture was proved also for all n ∈ Z and K⊗̂π A where A was only assumed to be a Banach C-algebra with one-sided bounded approximate identity [38]. Here we prove analogous theorems for K-theory of real Banach and C∗-algebras, and then deduce similar results in Hermitian K-theory. The article is organized as follows. In Chapter 1 we set the stage by introducing the concept of a K-ring, which is a slight generalization and a modification to what was called a “stable” algebra in [23]. We also introduce a novel notion of a stable retract. Then we proceed to demonstrate that the comparison map between algebraic and topological K-groups in degrees n ≤ 0 is an isomorphism for Banach algebras that are stable retracts of K⊗max A for some C∗-algebra A, or of K⊗̂π A, for some Banach algebra A (Theorem 1.1). We recall how to endow K∗(K) with a canonical structure of a Z-graded, associative, graded commutative, and unital ring in Chapter 2. If a K-ring is Hunital as a Q-algebra, we equip its Z-graded algebraic K-groups with a structure of a Z-graded unitary K∗(K)-module. We distinguish two cases: KR-rings and 1991 Mathematics Subject Classification. 19K99.